Without a photo and just your description to go by, this sounds like it could be coral spot fungus. However, coral spot (Nectria cinnibarina) typically only infects dead wood. Are they on outer twigs that might have experienced some cold weather dieback?

You might want to do some additional research - try Googling under the botanical name.

There has been some dying back of fig trees in this area during recent winters.

I see many bright orange patches growing on my fruit tree branches. The are growing on the surface and can be scraped away, although I don't. They grow amongst the various lichens. I consider all of this sort of stuff completely harmless.

It depends entirely on what the orange patches are. Some are indeed lichens or similar organisms and as such, harmless. But if it IS coral spot fungus, that is symptomatic of a larger problem and NOT harmless. Although coral spot typically infects deadwood - like cold damaged stems - it can migrate into living tissue and if left untreated, can cause the entire plant to die. Figs are rather prone to coral spot, as are currants (Ribes sp.). It can also be an indicator that the tree is suffering from some other stressor

To the OP - research the problem online. If confirmed as coral spot (lots of good photos to illustrate), then you need to prune out the affected branches, cutting back beyond the visible signs at least an inch to the next growth node (don't leave stubs). Destroy the prunings and be sure to sterilize your pruners between cuts. Wouldn't hurt to do a dormant fungicidal spraying either.

Thank you all.
Online photos of coral spot fungus look pretty much like what I have on my fig. So thank you gardengal48. I will take care of it tomorrow. Thanks for the info and tips.